Graphs such that all minimum dominating sets intersect all maximally independent sets (Q2839679)

Motivated by a problem about the inverse domination number, \(DI\)-pathological graphs are introduced as the graphs in which every minimum dominating set intersects every maximally independent set. It is proved that if \(G\) is a connected \(DI\)-pathological graph with \(\gamma(G)=3\), then every vertex from a minimum dominating set \(D\) has at least two private neighbors in \(V(G)\setminus D\). Consequently the unique connected \(DI\)-pathological graph with \(\gamma(G)=3\) and with the least number of vertices and edges is found. Infinite families of the so-called exploded paths and exploded cycles are proved to be \(DI\)-pathological. (Explosion is also known in the literature as the X-join and was introduced in [\textit{G. Sabidussi}, Math. Z. 76, 385-401 (1961; Zbl 0109.16404)]).